4,858 research outputs found
Degree subtraction eigenvalues and energy of graphs
The degree subtraction matrix of a graph is introduced, whose -th entry is , where is the degree of a vertex in . If is a non-regular graph, then has exactly two nonzero eigenvalues which are purely imaginary. Eigenvalues of the degree subtraction matrices of a graph and of its complement are the same. The degree subtraction energy of is defined as the sum of absolute values of eigenvalues of and we express it in terms of the first Zagreb index
Graph-theory induced gravity and strongly-degenerate fermions in a self-consistent Einstein universe
We study UV-finite theory of induced gravity. We use scalar fields, Dirac
fields and vector fields as matter fields whose one-loop effects induce the
gravitational action. To obtain the mass spectrum which satisfies the
UV-finiteness condition, we use a graph-based construction of mass matrices.
The existence of a self-consistent static solution for an Einstein universe is
shown in the presence of degenerate fermion.Comment: 16pages, 1figur
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
String Breaking from Ladder Diagrams in SYM Theory
The AdS/CFT correspondence establishes a string representation for Wilson
loops in N=4 SYM theory at large N and large 't Hooft coupling. One of the
clearest manifestations of the stringy behaviour in Wilson loop correlators is
the string-breaking phase transition. It is shown that resummation of planar
diagrams without internal vertices predicts the strong-coupling phase transtion
in exactly the same setting in which it arises from the string representation.Comment: 15 pages, 5 figures; v2: misprint in eq. (3.9) corrected; v4:
treatment of inhomogeneous term in the Dyson equation modifie
Integrability of two-loop dilatation operator in gauge theories
We study the two-loop dilatation operator in the noncompact SL(2) sector of
QCD and supersymmetric Yang-Mills theories with N=1,2,4 supercharges. The
analysis is performed for Wilson operators built from three quark/gaugino
fields of the same helicity belonging to the fundamental/adjoint representation
of the SU(3)/SU(N_c) gauge group and involving an arbitrary number of covariant
derivatives projected onto the light-cone. To one-loop order, the dilatation
operator inherits the conformal symmetry of the classical theory and is given
in the multi-color limit by a local Hamiltonian of the Heisenberg magnet with
the spin operators being generators of the collinear subgroup of full
(super)conformal group. Starting from two loops, the dilatation operator
depends on the representation of the gauge group and, in addition, receives
corrections stemming from the violation of the conformal symmetry. We compute
its eigenspectrum and demonstrate that to two-loop order integrability survives
the conformal symmetry breaking in the aforementioned gauge theories, but it is
violated in QCD by the contribution of nonplanar diagrams. In SYM theories with
extended supersymmetry, the N-dependence of the two-loop dilatation operator
can be factorized (modulo an additive normalization constant) into a
multiplicative c-number. This property makes the eigenspectrum of the two-loop
dilatation operator alike in all gauge theories including the maximally
supersymmetric theory. Our analysis suggests that integrability is only tied to
the planar limit and it is sensitive neither to conformal symmetry nor
supersymmetry.Comment: 70 pages, 10 figure
Neighborhood properties of complex networks
A concept of neighborhood in complex networks is addressed based on the
criterion of the minimal number os steps to reach other vertices. This amounts
to, starting from a given network , generating a family of networks
such that, the vertices that are steps apart in
the original , are only 1 step apart in . The higher order
networks are generated using Boolean operations among the adjacency matrices
that represent . The families originated by the well known
linear and the Erd\"os-Renyi networks are found to be invariant, in the sense
that the spectra of are the same, up to finite size effects. A further
family originated from small world network is identified
- …